Not Homework. Proving question

[icystrike]

Homework Statement

Prove that:

$$(\frac{10^{n}}{10^{n}-1})^{10^n} = e$$

as n approaches inf.

Homework Equations

It is rather obvious that i can let 10^{n} be yet another variable

The Attempt at a Solution

I proved by assuming binomial dist as Poisson dist (it is interesting for me to use this to prove a continued fraction)

 

[Mark44]

Homework Statement

Prove that:

$$(\frac{10^{n}}{10^{n}-1})^{10^n} = e$$

as n approaches inf.

Homework Equations

It is rather obvious that i can let 10^{n} be yet another variable

The Attempt at a Solution

I proved by assuming binomial dist as Poisson dist (it is interesting for me to use this to prove a continued fraction)

You can simplify by letting u = 10ⁿ and working with this limit:

$$\lim_{u \to \infty}\left( \frac{u}{u - 1} \right)^u$$

The usual approach is to let y = (u/(u - 1))^u, and then take the natural log of both sides.

ln y = u ln(u/(u - 1)) = [ln(u/(u - 1))]/(1/u)

Now take the limit of both sides. Note that the expression above is the limit of the log of what you want.